Types of studies Retrospective: case-control Prospective: – Clinical trial observational study – Cohort study Cross-sectional: experimental study
Comparing two proportions Difference Relative risk Odds ratio – Odds defined as – For a 2*2 table, odds ratio – Another name: cross-product ratio
Properties of the Odds Ratio 0=<θ <, θ=1 means independence of X and Y the farther from 1.0, the stronger the association between X and Y. log θ is convenient and symmetric Suitable for all direction No change when any row/column multiplied by a constant.
Partial association in stratified 2*2 tables Experimental studies We hold other covariates constant to study the effect of X on Y. Observational studies Control for a possibly confounding variable Z Partial tables=>conditional association Marginal table
Marginal versus Conditional(cont.) Marginal conditional
Homogeneous Association For a 2*2*K table, homogeneous XY association defined as: A symmetric property: – Applies to any pair of variables viewed across the categories of the third. – No interaction between two variables in their effects on the other variable.
Homogeneous Association(cont.) Suppose: – X=smoking(yes, no) – Y=lung cancer(yes, no) – Z=age( 65) – And Age is an Effect Modifier
Extensions for i*j Tables For a 2*2 table Odds ratio An i*j table Odds ratios
For I*J tables (I-1)*(J-1) odds ratios describe any association All 1.0s means INDEPENDENCE! Three-way I*J*K tables, Homogeneous XY association means: any conditional odds ratio formed using two categories of X and Y each is the same at each category of Z.
Measures of Association Two kinds of variables: – Nominal variables – Ordinal variables Nominal variables: Set a measure for X and Y: – V(Y),V(Y|X) Proportional reduction:
Measures of variation Entropy: Goodman and Kruskal(1954) (tau) Lambda:
About Entropy Uncertainty coefficient: U=0=>INDEPENDENCE U=1=>π(j|i)=1 for each i, some j. Drawbacks: No intuition for such a proportional reduction.